Photoshop runs on you machine it has always worked that way. There is no full function online web bast Photoshop that I know of.
Creative Cloud us a subscription based Photoshop. Photoshop still run on you machine. To Keep your Photoshop installation activated and working you are required from time to time to sign in with You subscription User ID and Password over the internet. I think the minimum subscription available is a year subscription. There is a LR + PS Photographer subscription that runs $10 a month.
Perpetual CS6 I only need an internet connection once when you activate it. There may even be a way to activate it offline with Adobe help